We shall now prove Theorem REF .
We will follow [1]} and sketch how to obtain (REF ) and (), with the case of the wave equation being similar.
With \(V_\ell \)  as in (REF ), we claim the following bound:
\(\int _{-\infty }^\infty \langle \xi \rangle ^{-2} \big | e^{it\mathcal {H}_\ell } u(\xi )\big |^2\, d\xi \lesssim \langle t\rangle ^{-2} \ell ^{4} \int _{-\infty }^\infty \langle \xi \rangle ^{2} |u(\xi )|^2\, d\xi \) 
